An In-Depth Analytical Study of Switching States of Direct Torque Control Algorithm for Induction Motor over the Entire Speed Range

: In this paper, a full analysis of the Voltage Vectors (VVs) in DTC algorithm is presented. The analytical analysis showed that the application of specific VVs results in false switching states called Uncontrollable Angles (UCAs). A robust scheme that ensures the elimination of (UCAs) is proposed for medium and high speeds with (18) subsectors (SSs). Simulation results were obtained and validated by using MATLAB/Simulin


Introduction
Direct torque control (DTC) characterized by fast dynamic response, structural simplicity, and much simpler than the FOC [1][2][3]. Several improvements have been made to overcome problems associated with the DTC drive such as high torque ripple in particular [4]. Reference [5] provided an in-depth study of the VVs effect on the state variables issue over the entire speed range in terms of UnAs. To select the appropriate VV, a new approach is initially introduced by inserting the zero VV along with the selected one [6]. Research [7] eliminated the zero VVs during torque dynamics to establish a fast torque response in the transient state. A modified LUT for DTC of three-level dual VSI fed openended winding IM drive was proposed in [8], where the VVs selection for lower hysteresis boundary conditions were restructured with null voltages. Authors in [9] used the concept of virtual vectors for a seven-phase IM where the torque ripple in different operation conditions was investigated. A universal LUT is proposed for OW-PMSM drives under different conditions of the dc-link voltage ratio [10][11][12]. The proposed strategy effectively optimized the duty ratio of fundamental VV to minimize the error between reference VV and final VV imposed on motor terminals. A duty-ratio regulator that considers the operating speed impact on the torque deviation of the active voltage vectors was proposed in [12]. This article suggests an enhanced, simple, and effective DTFRC strategy that aims to eliminate the UnAs over the wide speed range. The proposed method, which uses (18) SSs for the rotor at medium and high speeds, overcomes the conventional (6)

Analytical Modeling
The basic principle of DTRFC is summed up in the instantaneous control of both rotor flux and the torque using intermediate loops without current control ones. the two components of the rotor flux vector of the rotor are estimated in the stator reference frame (α sβ s ) as the following: where Lm is the mutual inductance. Ls, and Lr are the stator and rotor self inductance respectively. σ is the leakage factor. The rotor flux vector will be oriented according to α-coordinate axis. Thus, the imaginary component of the rotor flux vector will be zero, i.e.,: Φrβ=0, Φr= Φrα. The derivatives of two controlled variables (Φr, Tem) are known as [5]: where k Φ r is an optional positive constant. Depending on the previous equations, the block diagram of the DTRFC algorithm can be constructed.
To enhance the performance at medium and high speeds, a transition will be allowed between the conventional strategy with (6) sectors for the low-speed range, and the improved strategy with (18) SSs (for medium and high speeds) as shown in Figure 1.

Determination of the UnAs Values for Low and High Speeds
Depending on the equations (2) and (3), the effect of applying the two vectors Vi+1, Vi−1 on dSΦr at low speed (20% ωn) and Vi+2, Vi−2 at high speed were analyzed. There are two UCAs at low speed and high speed with value of each π/69[rad/s] and π/94[rad/s], Respectively. In addition, the analysis is achieved for the two VVs (Vi+1, Vi+2) on dSTem at high speed. There are two UCAs. Each of them has a value equals π/13[rad/s] (22%) of the sector. Table 1 summarizes the values of the UnAs for each (dSΦr, dSTem) over the entire speed range.

The Improved Strategy (18) SSs DTRFC Strategy for Medium-High Speeds
The proposed strategy is based on dividing the path of the rotor flux into (18) unequal SSs. Every three subsequent sectors will repeat the same distance after the previous three SSs. Depending on Equations (2) and (3), the position of the error change for the rotor flux and torque for a high speed (75% ωn) were analyzed in order to devise the improved lookup table. The LUT for the improved strategy is shown in Table 2.
For the rest of the SSs, the applied vectors can be known by increasing the vector index by (1) when moving between SSs according to the following sequence:

Determination of the Transition Speed ωT between the Traditional and the Proposed Strategy
It is important to determine the speed at which UnAs start to appear, i.e., the transition speed ωT. An increment was given to the angle θΦr, so it scans the entire sector. The speed was given a value starting from zero within an iterative loop during which the two components (Vsα, Vsβ) were calculated. The derivatives (dSΦr, dSTem) were calculated according to the two equations (2&3). The speed range at which UnAs disappeared were within (0:55% ωn), or (0:155[rad/s]). However, if the speed exceeds (155/rad/s), UnAs start to appear.

Simulation Results and Discussion
By simulating the proposed block diagram, in MATALB/Simulink environment, the simulation results are obtained in Figure 2

Conclusions
This paper provided an analytical investigation of the DTRFC algorithm over the entire speed range in terms of UnAs. The proposed scheme with (18) SSs is devised aimed to eliminate the UnAs of some VVs for medium and high speeds which yields a correct torque response. The transition speed value, at which the UnAs begin to appear, has been precisely analytically determined. Furthermore, the proposed method combined the advantages of conventional and improved strategies to work over a wide speed range. The simulation results validated the feasibility and effectiveness of the proposed scheme in IM drives over the wide speed range.
Artificial network techniques for the transition state between the two algorithms is the main goal for the future work.

Conflicts of Interest:
The authors declare no conflict of interest.
Appendix A   Table A1. Parameters of three phase induction motor.